Proof. This theorem is a corollary of the (much more general) Kruskal-Katona theorem. The Kruskal-Katona theorem has a very hands-on proof, based on iteratively modifying the graph. We will see a linear-algebraic proof. Let A be the n×n adjacency matrix of G (Auv = 1 if vertex u is adjacent to vertex v, and Auv = 0 otherwise). Note that A is symmetric. It turns out that e and t are both fundame...

Calculating the statistical linear response of turbulent dynamical systems to the change in external forcing is a problem of wide contemporary interest. Here the authors apply linear regression models with memory, AR(p) models, to approximate this statistical linear response by directly fitting the autocorrelations of the underlying turbulent dynamical system without further computational exper...

This chapter concerns rank tests for independence of two random variables. A form of locally most powerful test against particular alternative is derived and the most often used variants of this test are mentioned, namely the test of van der Waerden type, Spearman rank correlation coefficient, the quadrant test and the Kendall rank correlation coefficient, a member of non-linear rank statistics...